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complex numbers - winding number

Hi all,

I need to prove the following lemma in attachment . Our instructor wants us to do it in the following way:

Let A be the set of all $\displaystyle x \in [0,1] $ with the property that there exists a continuous function

$\displaystyle \theta : [0,x] -> R $ such that $\displaystyle \theta (0) = \theta_{0} $

and

$\displaystyle \gamma (t) = | \gamma (t) | e^{i \theta (t) } t \in [0,x] $

Prove that A is nonempty , closed and relatively open in [0,1] and conclude that A = [0,1] since [0,1] is connected.

Can anyone help?? I don't even have a slight idea how to proceed.